Coherent structures in wall turbulence
Smallest scale of the flow: kolmogorov scale
(in the near atmosphere about 1mm)
Largest scale of the flow: several times the boundary layer height
(in the atmosphere may go up to O(1-10 Km )
There are 6-7 orders of magnitude !
However IF, we understand how turbulent structures behave and IF
these structures truly play a major role (statistically) on momentum,
scalar and energy fluxes, mixing, etc. ...
Then we could propose low dimensional models, smart closures,
control systems
Short term goal: understand and control near wall processes
(relevant for drag, lift, resuspension, etc)
Long term goal: shift turbulent closure to larger scales, in order to solve
large domain accurately (atmosphere, rivers, oceans)
Flow visualization and sketches
Kline 1967 (near wall streaks)
(log and outer layer)
(1981)
Acarlar and Smith, 1987, downstream of a fixed hemisphere
downstream of a low momentum
fluid ejection
Laminar flow upstream
Kline 1967
Flow visualization (hydrogen bubbles, flow markers)
Robinson 1991
Hairpin vortex detection: track of a strongly 3D structure on the
2D streamwise - wall normal laser sheet: Adrian et al 2000
The Biot-Savart law is used to calculate the velocity
induced by vortex lines.
For a vortex line of infinite length, the induced velocity
at a point is given by:
V = 2 πΓ /d where
Γ is the strength of the vortex
d is the shortest distance from a point P to the
vortex line
For a arch-like vortex line, there is a combined
induction towards its center
(ejection of low momentum fluid u’v’ Q2 event
Q2 event
vortex
Shear layer
A brief summary . . .
• Single hairpin vortices can explain the observed
features of low and high speed streaks, bursting
phenomena and lift up of structures (viscous &
buffer layer)
• What is still missing so far is the outer layer,
• Structures were observed to form bulges with
ramp-like features.
Numerical Simulation (Zhou, Adrian et al. 1996, 1999)
isovorticity surface
Self sustaining mechanism
(see also Waleffe 1990) and
vortex alignment
Limitation : low Re
with initial perturbation
Experimental evidence of hairpin packets
in smooth wall turbulence
(Adrian, Meinhart, Tomkins JFM, 2000)
Instantaneous flow fields: U-Uc (convection velocity)
Vortex marker: swirling strength
Q4
Q4
Ramp packet
Q2
Detection of zones of uniform
momentum associated to the
streamwise alignment of
hairpin vortex: mutual
induction of Q2 event
Vortex identification
Okubo-Weiss parameter
Q  S 2   z2
S 2  Sn2  Ss2
where :
Sn  ux  wz
Ss  wx  uz
From the local velocity
gradient tensor
u
 x
u 
w
x
Swirling Strength analysis
u
z
w
z
See also Chong & Perry, 1990
Jeong and Hussain 1995
Imaginary eigenvalues
c  cr  ici
We select the region
where
ci  0
Numerical simulation Adrian, PoF 2008 multimedia appendix
Flow visualization,
Statistical Signature
1)Relevance
2)Physical mechanisms
3)Connection with quadrant analysis (Lu &
Willmart, 1973, Wallace 1972, Nezu &
Nakagava 1977)
4)Vortex identification in 2D and 3D
5)Zones of uniform momentum
6)Consistency with observed resuspension
events (strong correlation between c’w’ and u’w’ events)
Besides instantaneous realizations…
Is it possible to obtain some quantitative
information about turbulent structures ?
2 point correlation
2 point correlation tensor
Rij rx , y, y '  ui  x, y  u *j  x  rx , y '
correlation coefficient
(normalized)
Rij rx , y, y '
ρ ij 
for i, j  u, v
σ i y σ j y'
See also Proper
Orthogonal
Decomposition
(Holmes & Lumley )
Linear stochastic estimate
Estimate of the flow field
Statistically conditioned
To the realization of a
known event :
1) II quadrant (u < 0, v > 0)
2) IV quadrant (u > 0, v < 0)
3) Vortex
identified by the swirling strength 
complex part of the eigenvalue of
the local velocity gradient tensor.
Comparison A B center
(reduction of the streamwise lengthscale:
lost of coherence within the structures
of the packets)
(see also Krogstad e Antonia 1994 rough wall)
A
B
Two point correlation
streamwise velocity fluctuation
Comparison A B center
A
B
Linear Stochastic Estimate:
Question:
What is the average flow field statistically conditioned to the realization of a
vortex with a spanwise axe (identified as the signature of the hairpin vortex
On the laser sheet)?
The best (linear) estimate is given by
  

 u j x' 
λ
x
 


u j x' λx   L jλx  
  λ x 
λ x  λ  x 

con x  ( x, y )
  
λx  u j x'  λx, y' u x  rx , y 
Adrian, Moin & Moser, 1987
Adrian 1988, Christensen 2000
Note:
Information about conditioned
probabilistic variables are obtained from
unconditioned statistical moments
Linear stochastic
Estimate :
E
known event
assumed at a fixed y’
See Christensen 2000
Flow field obtained from a statistical
analysis (conditioned to the realization of
a E event)
E
Kim and Adrian 1999
Spanwise alignment of hairpin structures leading to
long coherent regions of uniform momentum
Kim & Adrian 1999
VLSM Contribution :
turbulent kinetic energy and Reynolds stresses
Guala et al. 06
Net force exerted by Reynold stress in the mean momentum equation
• Pipe flow
• Turbulent Boundary layers
• channel flow
Pipe : Guala et al, 06
channel:
Turb. B. layer: Balakumar, (2007)
A brief summary . . .
• Large scale motion participate significantly to the
Reynolds stress, thus contribute not only to TKE
but also to TKE production.
• In terms of momentum balance, close to the wall,
VLSM push the flow forward, while smaller scales
slow down the flow.
• Such features are observed for turbulent pipe,
channel and boundary layers flows
Marusic & Hutchins 2008
Atmospheric Surface Layer
Reτ=O(106)
Hutchins & Marusic 2007
High Re
Large scale influence on the near
Wall turbulence intensity:
Amplitude modulation
Low Re
High Re
Low Re
Note that in different research field
some type of very large scale
motions are addressed with
different names
e.g. streamwise rolls (atmospheric
science) or secondary current
(river hydraulics)
VLSM : A visual inspection
PIPE FLOW
Lekakis 88,
ATMOSPHERIC
SURFACE LAYER (ASL)
Metzger et al. 07; Guala, Metzger, McKeon 08
Chacin & Cantwell 2000 (Turb. Boundary Layer)
Soria 94
Chong & Perry 90
A different view
Luthi 2005
PTV isotropic turbulence
Coherent structures vs vortices
Open questions:
1) How spanwise mean vorticity relates to streamwise fluctuating vorticity ?
2) How vortex stretching is affected by a non zero mean strain
( and perhaps also mean vorticity) ?
3) Do they both scale with Kolmogorov (core) and the integral lengthscale ?
4) Are they more or less stable as compared to worms in isotropic 3D turbulence ?
Other Questions
1) How roughness in general can perturb self organization, how about complex
terrain ?
2) What are the relevant scales for coherent structures (inner, outer)?
3) Can we really define a coherent structure
4) Can we describe coherent structures evolution in
unambiguous quantitative (not handwavy) terms ?
5) How VLSM rtelate to hairpin packets (is it Reynolds dependent)?
6) why near wall peak can be affected by outer layer structures?
7) which terms of which equation are responsible?
8) can we go beyond geometrical characteristics (exp) and vorticity contour (num) ?